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About 987 papers for “Cartesian tensor”

Cartesian Tensor

Bipin Singh Koranga, Sanjay Kumar Padaliya · River Publishers eBooks · 2022

Cited by 0

Cartesian Tensor Analysis

Cambridge University Press eBooks · 2009

In this chapter we present an elementary introduction to Cartesian tensor analysis in a three-dimensional Euclidean point space or a two-dimensional subspace. A Euclidean point space is the space of position vectors of points. The term vector is used in the sense of classical vector analysis, and scalars and polar vectors are zeroth- and first-order tensors, respectively. The distinction between polar and axial vectors is discussed later in this chapter. A scalar is a single quantity that possesses magnitude and does not depend on any particular coordinate system, and a vector is a quantity that possesses both magnitude and direction and has components, with respect to a particular coordinate system, which transform in a definite manner under change of coordinate system. Also vectors obey the parallelogram law of addition. There are quantities that possess both magnitude and direction but are not vectors, for example, the angle of finite rotation of a rigid body about a fixed axis.

Cited by 0

Summary of Cartesian tensor notation

Samuel Paolucci · 2016

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Cited by 0

CarNet - Cartesian Natural Tensor Networks

Wen, Mingjian · Zenodo (CERN European Organization for Nuclear Research) · 2025

======================================================================================= The codes provided here is a copy of the carnet repo hosted on GitHub, which will be made publicly available shortly. ======================================================================================= CarNet Cartesian Natural Tensor Networks - atomistic machine learning for interatomic potentials, scalar properties, tensorial properties and beyond. Installation Dependencies Python >= 3.10 PyTorch >= 2.5 Installing CarNet Download this zip file, and unzip it. Then install by, $ cd carnet $ pip install -e . If you get into trouble with some of the dependencies, try $ pip install -e '.[strict]' which will install the dependencies with pinned versions. Model Training Example training scripts and configuration files can be found in the scripts directory. For example, a toy example for training a CarNet potential on a small dataset of water molecules can be done via: python train_ip.py You will need to adjust configs/config_ip.yaml to change trainset_path, valset_path, and testset_path to point to the correct dataset location. Pretrained models Pretrained models are stored in the pretrained_models directory, including: MLIPs for bulk LiPS and water, and molecular ethanol Tensor property prediction models for ethanol Elastic tensor prediction models for inorganic crystals Scripts to use the pretrained models for making predictions can be found in the scripts directory. Molecular dynamics simulations Scripts for running molecular dynamics simulations with CarNet potentials using ASEcan be found in the examples directory.

Cited by 0Open access

CarNet - Cartesian Natural Tensor Networks

Wen, Mingjian · Zenodo (CERN European Organization for Nuclear Research) · 2025

======================================================================================= The codes provided here is a copy of the carnet repo hosted on GitHub, which will be made publicly available shortly. ======================================================================================= CarNet Cartesian Natural Tensor Networks - atomistic machine learning for interatomic potentials, scalar properties, tensorial properties and beyond. Installation Dependencies Python >= 3.10 PyTorch >= 2.5 Installing CarNet Download this zip file, and unzip it. Then install by, $ cd carnet $ pip install -e . If you get into trouble with some of the dependencies, try $ pip install -e '.[strict]' which will install the dependencies with pinned versions. Model Training Example training scripts and configuration files can be found in the scripts directory. For example, a toy example for training a CarNet potential on a small dataset of water molecules can be done via: python train_ip.py You will need to adjust configs/config_ip.yaml to change trainset_path, valset_path, and testset_path to point to the correct dataset location. Pretrained models Pretrained models are stored in the pretrained_models directory, including: MLIPs for bulk LiPS and water, and molecular ethanol Tensor property prediction models for ethanol Elastic tensor prediction models for inorganic crystals Scripts to use the pretrained models for making predictions can be found in the scripts directory. Molecular dynamics simulations Scripts for running molecular dynamics simulations with CarNet potentials using ASEcan be found in the examples directory.

Cited by 0Open access

Linear invariants of a Cartesian tensor

Faiz Ahmad, M. A. Rashid · The Quarterly Journal of Mechanics and Applied Mathematics · 2008

The number of linear invariants under SO(3) as well as SO(2) of a Cartesian tensor of an arbitrary rank is studied. A linear form is defined in terms of elements of a tensor. It is established that the number of linear invariants of a tensor of rank n under SO(3) equals the dimension of the space of isotropic tensors of rank n. Formulas for the number of invariants in the two cases are also derived. For the elasticity tensor, our analysis confirms the results of Norris.

Cited by 3Open access

Cartesian-nj: Extending e3nn to Irreducible Cartesian Tensor Product and Contracion

Zemin Xu, Chenyu Wu, Wenbo Xie et al. · ArXiv.org · 2025

Equivariant atomistic machine learning models have brought substantial gains in both extrapolation capability and predictive accuracy. Depending on the basis of the space, two distinct types of irreducible representations are utilized. From architectures built upon spherical tensors (STs) to more recent formulations employing irreducible Cartesian tensors (ICTs), STs have remained dominant owing to their compactness, elegance, and theoretical completeness. Nevertheless, questions have persisted regarding whether ST constructions are the only viable design principle, motivating continued development of Cartesian networks. In this work, we introduce the Cartesian-3j and Cartesian-nj symbol, which serve as direct analogues of the Wigner-3j and Wigner-nj symbol defined for tensor coupling. These coefficients enable the combination of any two ICTs into a new ICT. Building on this foundation, we extend e3nn to support irreducible Cartesian tensor product, and we release the resulting Python package as cartnn. Within this framework, we implement Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first systematic comparison of Cartesian and spherical models to assess whether Cartesian formulations may offer advantages under specific conditions. Using TACE as a representative example, we further examine whether architectures constructed from irreducible Cartesian tensor product and contraction(ICTP and ICTC) are conceptually well-founded in Cartesian space and whether opportunities remain for improving their design.

Cited by 0Open access

Cartesian-nj: Extending e3nn to Irreducible Cartesian Tensor Product and Contracion

Zemin Xu, Chenyu Wu, Wenbo Xie et al. · arXiv (Cornell University) · 2025

Equivariant atomistic machine learning models have brought substantial gains in both extrapolation capability and predictive accuracy. Depending on the basis of the space, two distinct types of irreducible representations are utilized. From architectures built upon spherical tensors (STs) to more recent formulations employing irreducible Cartesian tensors (ICTs), STs have remained dominant owing to their compactness, elegance, and theoretical completeness. Nevertheless, questions have persisted regarding whether ST constructions are the only viable design principle, motivating continued development of Cartesian networks. In this work, we introduce the Cartesian-3j and Cartesian-nj symbol, which serve as direct analogues of the Wigner-3j and Wigner-nj symbol defined for tensor coupling. These coefficients enable the combination of any two ICTs into a new ICT. Building on this foundation, we extend e3nn to support irreducible Cartesian tensor product, and we release the resulting Python package as cartnn. Within this framework, we implement Cartesian counterparts of MACE, NequIP, and Allegro, allowing the first systematic comparison of Cartesian and spherical models to assess whether Cartesian formulations may offer advantages under specific conditions. Using TACE as a representative example, we further examine whether architectures constructed from irreducible Cartesian tensor product and contraction(ICTP and ICTC) are conceptually well-founded in Cartesian space and whether opportunities remain for improving their design.

Cited by 0Open access

Cartesian Tensor Perturbation Scheme for the Senftleben Effect

Eugene R. Cooper, David K. Hoffman · The Journal of Chemical Physics · 1971

A perturbation expansion scheme using a Cartesian tensor expansion is developed for computing transport coefficients of diatomic gases in a static magnetic field. The collision operator is uniquely separated into an unperturbed part plus a “nonspherical” perturbation. Special attention is given to the saturation limit. Using a truncated expansion set, explicit expressions for the thermal conductivity and viscosity are obtained to second order in the “nonsphericity.” These solutions in the second-order approximation are identical with the approximate solutions of Levi and McCourt. Also, the fourth-order contributions from terms odd in the angular momentum are given in the saturation limit, since for certain molecular models the second-order contributions are zero.

Cited by 13

Invariants of a Cartesian tensor of rank 3

Faiz Ahmad · SHILAP Revista de lepidopterología · 2010

General methods are applied to find complete sets of invariants of a tensor of rank 3. When the results are specialized to the piezoelectric tensor, it is found that the tensor has no linear invariant. Also under SO(2), as well as SO(3), the piezoelectric tensor has five quadratic invariants. The sets of invariants are complete.

Cited by 10Open access

CARTESIAN TENSOR EXPANSION OF THE FOKKER–PLANCK EQUATION

I. P. Shkarofsky · Canadian Journal of Physics · 1963

The Fokker–Planck equation is expanded via Cartesian tensors. The results, valid for arbitrary mass ratio, reduce to known expressions for the f 0 , (isotropic) and f 1 (directional) parts of the distribution. The same analysis also gives the relation for the f 2 (tensor) part of the distribution. Nonlinear and recoil terms are also investigated. The f 0 relation yields relaxation times and energy equipartition times for arbitrary velocity distributions. The ion recoil term is important for low-frequency waves in the electron f 1 equation but is negligible in the electron f 2 equation. The electron–electron contributions to the f 1 and f 2 collisional terms are of the same order as the electron–ion contributions. Except for momentum transfer calculations, the ion–ion collisional terms dominate over the ion–electron terms in the ion collision equations referring to the directional (F 1 ) and tensor (F 2 ) parts of the ion distribution.

Cited by 36